A001159 sigma_4(n): sum of 4th powers of divisors of n.

1, 17, 82, 273, 626, 1394, 2402, 4369, 6643, 10642, 14642, 22386, 28562, 40834, 51332, 69905, 83522, 112931, 130322, 170898, 196964, 248914, 279842, 358258, 391251, 485554, 538084, 655746, 707282, 872644, 923522, 1118481, 1200644

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• mult

Comments

If the canonical factorization of n into prime powers is the product of p^e(p) then sigma_k(n) = Product_p ((p^((e(p)+1)*k))-1)/(p^k-1).

Sum_{d|n} 1/d^k is equal to sigma_k(n)/n^k. So sequences A017665-A017712 also give the numerators and denominators of sigma_k(n)/n^k for k = 1..24. The power sums sigma_k(n) are in sequences A000203 (k=1), A001157-A001160 (k=2,3,4,5), A013954-A013972 for k = 6,7,...,24. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 05 2001

sigma_4(n) is the sum of the 4th powers of the divisors of n (A001159).

References

• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972, p. 827.
• T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 38.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972, p. 827.
• Index entries for sequences related to sigma(n)

Crossrefs

Cf. A000005, A000203, A001157, A001158.

Programs

Magma

[DivisorSigma(4,n): n in [1..40]]; // Bruno Berselli, Apr 10 2013

Maple

with(numtheory); A001159 := proc(n) sigma[4](n) ; end proc: # R. J. Mathar, Feb 04 2011

Mathematica

lst={}; Do[AppendTo[lst, DivisorSigma[4,n]], {n,5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Mar 11 2009 *)
DivisorSigma[4,Range[40]] (* Harvey P. Dale, Apr 28 2013 *)

Maxima

makelist(divsum(n,4),n,1,100); /* Emanuele Munarini, Mar 26 2011 */

PARI

N=99;q='q+O('q^N);
Vec(sum(n=1,N,n^4*q^n/(1-q^n))) /* Joerg Arndt, Feb 04 2011 */

Sage

[sigma(n,4)for n in range(1,34)] # Zerinvary Lajos_, Jun 04 2009

Formula

Multiplicative with a(p^e) = (p^(4e+4)-1)/(p^4-1). - David W. Wilson, Aug 01 2001

G.f. Sum_{k>=1} k^4*x^k/(1-x^k). - Benoit Cloitre, Apr 21 2003

L.g.f.: -log(Product_{j>=1} (1-x^j)^(j^3)) = Sum_{n>=1} a(n)/n*x^n. - Joerg Arndt, Feb 04 2011

Dirichlet g.f.: zeta(s)*zeta(s-4). - R. J. Mathar, Feb 04 2011

a(n) = Sum_{d|n} tau_{-2}^(d)*J_4(n/d), where tau_{-2} is A007427 and J_4 A059377. - Enrique Pérez Herrero, Jan 19 2013

G..f.: Sum_{n >= 1} A(4,x^n)/(1 - x^n)^5, where A(4,x) = x + 11*x^2 + 11*x^3 + x^4 is the 4th Eulerian polynomial - see A008292. - Peter Bala, Jan 11 2021

a(n) = Sum_{1 <= i, j, k, l <= n} tau(gcd(i, j, k, l, n)) = Sum_{d divides n} tau(d) * J_4(n/d), where the divisor function tau(n) = A000005(n) and the Jordan totient function J_4(n) = A059377(n). - Peter Bala, Jan 22 2024